For more than two decades, a single model for the spreading of a surfactant-driven thin liquid film has dominated the applied mathematics literature on the subject. Recently, through the use of fluorescently-tagged lipids, it has become possible to make direct, quantitative comparisons between experiments and models. These comparisons have revealed two important discrepancies between simulations and experiments: the spatial distribution of the surfactant layer, and the timescale over which spreading occurs. In this paper, we present numerical simulations that demonstrate the impact of the particular choice of the equation of state (EoS) relating the surfactant concentration to the surface tension. Previous choices of the model EoS have been an ad-hoc decreasing function. Here, we instead propose an empirically-motivated equation of state; this provides a route to resolving some discrepancies and raises new issues to be pursued in future experiments. In addition, we test the influence of the choice of initial conditions and values for the non-dimensional groups. We demonstrate that the choice of EoS improves the agreement in surfactant distribution morphology between simulations and experiments, and impacts the dynamics of the simulations. The relevant feature of the EoS, the gradient, has distinct regions for empirically motivated choices, which suggests that future work will need to consider more than one timescale. We observe that the non-dimensional number controlling the relative importance of gravitational vs. capillary forces has a larger impact on the dynamics than the other non-dimensional groups. Finally, we observe that the experimental approach of using a ring to contain the surfactant could affect the surfactant and fluid dynamics if it disrupts the intended initial surfactant distribution. However, the meniscus itself does not significantly affect the dynamics.
当表面活性分子在液体界面释放时，它们的扩散动力学由Marangoni流控制。虽然这样的马兰戈尼蔓延调查在不同的限制，精确的解决方案仍然很少。在这里，我们考虑沿深流体层的界面的不溶性表面活性剂的蔓延。对于二维Stokes流，最近表明，非线性输运问题可以精确地映射到复杂的Burgers方程[Crowdy，SIAM J. Appl. Math. 81，2526（2021）]。我们首先提出一个非常简单的推导这个方程。然后，我们提供了完全明确的解决方案，并发现，不同的初始表面活性剂分布-脉冲，孔，或周期性的-结果在不同的传播行为。在得到基本解的基础上，我们还讨论了表面扩散的影响。我们确定的情况下，传播可以被描述为一个有效的扩散过程，但观察到这种近似是不普遍有效的。最后，简要地考虑了具有轴对称的三维流动的情况。我们的研究结果应该提供参考解决方案的马兰戈尼蔓延，可以用荧光或光开关表面活性剂进行实验测试。
